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Three Card 2nd Chance Poker

Introduction

Three Card 2nd Chance is a poker-based game, invented and patented by me (patent number ). The "Best Hand Bonus" side bet was the idea of WagerWorks. Three Card 2nd Chance follows the same structure as a five-card version, Five-Card Mulligan Poker, which has been available at Odds On Internet casinos since May 2007. The thrust of the game is that if the player doesn't like his first hand, he may exchange it for a second hand by merely increasing his bet. The dealer also has the ability to replace his hand in certain circumstances. Then, the higher hand wins, and all wins pay at least even money. WagerWorks has an exclusive license to be the sole provider of Three Card 2nd Chance for Internet-based casinos. I would be happy to hear from "brick and mortar" casinos interested in offering the game.

Rules

A single 52-card deck is used. All hands are scored according to conventional poker rules, except the hand order is altered for 3-card hands.

Play starts with the player making an Ante bet. The player may optionally make a "Best Hand Bonus" bet as well. The most the player may bet on the Best Hand Bonus is two times the Ante bet.

The dealer shall give the player three cards and herself three cards. Player cards will be face up and dealer cards face down.

The player has the choice to either stand or to trade in his hand for a new three-card hand.

If the player chooses to trade, then he should discard his cards and make a Raise wager equal to the Ante wager.

The dealer will then turn over his cards.

If the dealer has a king high or higher, he will stand.

If the dealer has a queen high or less, he will discard his hand for three new cards.

The dealer will then compare her own hand to the player's hand, and the higher hand wins. If the dealer has the higher hand, then the player will lose his Ante wager as well as any Raise wager. If the player has the higher hand, then both Ante wager and Raise wager will be paid according to the Ante and Raise pay table below. This table also reflects the hand ranking order. A tie will result in a push.

Ante and Raise Pay Table

Hand

Pays

Straight flush

6 to 1

Three of a kind

4 to 1

Straight

3 to 2

All other

1 to 1

The Best Hand Bonus wager shall pay according to the higher of the player's and dealer's final hands, as follows.

Best Hand Bonus Pay Table

Hand

Pays

Three aces

35 to 1

Straight flush

15 to 1

Three of a kind

12 to 1

Straight

3 to 1

Flush

2 to 1

All others

Loss

Screenshot courtesy of WagerWorks

Strategy

If the player makes the ante wager only, then the optimal strategy is to switch with K87 or less, and stand on K92 or more.

If the player makes the maximum Best Hand Bonus bet, then the optimal strategy is to switch with AQJ or less, and stand with AK2 or more.

Of the two bet mixes, the one that results in the lower overall house edge is to make the maximum Best Hand Bonus bet.

Ante Only Analysis

The next table shows the possible outcomes if the player bets the Ante only, switching with K87 or less.

Ante Only Return Table — Switch with K87 or Less

Event

Bet

Pays

Probability

Return

Player wins with straight flush

2

12

0.001041

0.012497

Player wins with three of a kind

2

8

0.001125

0.009002

Player wins with straight flush

1

6

0.002176

0.013055

Player wins with three of a kind

1

4

0.002349

0.009395

Player wins with straight

2

3

0.015199

0.045598

Player wins with straight

1

1.5

0.031641

0.047461

Player wins with flush

2

2

0.021846

0.043693

Player wins with pair

2

2

0.061861

0.123722

Player wins with nothing

2

2

0.080141

0.160282

Player wins with flush

1

1

0.045172

0.045172

Player wins with pair

1

1

0.127912

0.127912

Player wins with nothing

1

1

0.117501

0.117501

Player ties

2

0

0.000482

0

Player ties

1

0

0.000637

0

Player loses

1

-1

0.189183

-0.189183

Player loses

2

-2

0.301734

-0.603468

Total

1

-0.03736

The lower right cell shows a house edge of 3.736%. The player will raise 48.34% of the time, for an average final wager of 1.4834 units. That makes theElement of Risk 3.376%/1.4834 = 2.518%.

2 to 1 Ratio of Side Bet to Ante Analysis

When the player makes the Best Hand Bonus bet, he should be more aggressive about switching. The following table shows the outcome of the Ante wager when the player switches on AQJ or less, the optimal strategy when making the maximum Best Hand Bonus side bet.

Ante Only Return Table — Switch with AQJ or Less

Event

Bet

Pays

Probability

Return

Player wins with straight flush

2

12

0.001553

0.018641

Player wins with three of a kind

2

8

0.001669

0.013354

Player wins with straight flush

1

6

0.002174

0.013046

Player wins with three of a kind

1

4

0.002345

0.009379

Player wins with straight

2

3

0.022650

0.067950

Player wins with flush

2

2

0.032376

0.064751

Player wins with straight

1

1.5

0.031611

0.047416

Player wins with pair

2

2

0.091599

0.183197

Player wins with nothing

2

2

0.118533

0.237066

Player wins with flush

1

1

0.045192

0.045192

Player wins with pair

1

1

0.127938

0.127938

Player wins with nothing

1

1

0.016185

0.016185

Player ties

1

0

0.000170

0

Player ties

2

0

0.000704

0

Player loses

1

-1

0.057368

-0.057368

Player loses

2

-2

0.447933

-0.895867

Total

1

-0.109119

The next table shows the possible outcomes of the Best Hand Bonus bet, when the player switches with AQJ or less.

Best Hand Bonus Return Table — Switch with AQJ or Less

Event

Pays

Probability

Return

Three aces

35

0.000569

0.019902

Straight flush

15

0.006811

0.102171

Three 2-K

12

0.006783

0.081396

Straight

3

0.099023

0.297068

Flush

2

0.140714

0.281428

Loser

-1

0.746100

-0.746100

Total

1

0.035865

The Ante wager table shows a player return of -10.912%. The Best Hand Bonus table shows a player return of 3.587%. Taking a weighted average, the overall player return is (1/3)×-0.109119 + (2/3)×0.035865 = -0.012463. So, the combined house edge is 1.246%. The player will raise 71.7% of the time, for an Element of Risk of 1.006%.

Methodology

This analysis was done by random simulation. With the possibility of both player and dealer switching cards, the number of possible combinations is combin(52,3) × combin(49,3) × combin(46,3) × combin(43,3) = 76,277,828,779,152,000, which would take years to loop through.